computational cell biology - christopher fall, eric marland, john wagner, john tyson

Marsden Department of Mathematics Control and Dynamical Systems Institute for Physical Science and Technology California Institute of Technology University of Maryland Pasadena, CA 91125

Christopher P Fall

SPRINGER

Interdisciplinary Applied Mathematics

tween the disciplines has led to the establishment of the series: Interdisciplinary Applied

Mathe-matics.

The purpose of this series is to meet the current and future needs for the interaction betweenvarious science and technology areas on the one hand and mathematics on the other This is done,firstly, by encouraging the ways that mathematics may be applied in traditional areas, as well aspoint towards new and innovative areas of applications; and, secondly, by encouraging otherscientific disciplines to engage in a dialog with mathematicians outlining their problems to bothaccess new methods and suggest innovative developments within mathematics itself

The series will consist of monographs and high-level texts from researchers working on the play between mathematics and other fields of science and technology

inter-Volumes published are listed at the end of this book.

Christopher P Fall Eric S Marland John M Wagner John J Tyson

Editors

Computational Cell Biology

With 210 Illustrations

New York University Appalachian State University Technology

New York, NY 10003 Boone, NC 28608 University of Connecticut Health

fall@cns.nyu.edu marlandes@appstate.edu Farmington, CT 06030

USAjwagner@nso.uchc.eduJohn J Tyson

S.S Antman J.E Marsden

Department of Mathematics Control and Dynamical Systems

Institute for Physical Science and Technology California Institute of Technology

University of Maryland Pasadena, CA 91125

College Park, MD 20742 USA

USA

Division of Control and Dynamical Systems

Applied Mathematics Mail Code 107-81

Brown University California Institute of Technology

Providence, RI 02912 Pasadena, CA 91125

Mathematics Subject Classification (2000): 92-01, 92BXX, 92C30, 92C20

Library of Congress Cataloging-in-Publication Data

Computational cell biology / editors, Christopher P Fall [et al.].

p cm — (Interdisciplinary applied mathematics)

Includes bibliographical references and index.

ISBN 0-387-95369-8 (alk paper)

1 Cytology—Computer simulation 2 Cytology—Mathematical models I Fall, Christopher P.

II Series.

QH585.5.D38 C65 2002

ISBN 0-387-95369-8 Printed on acid-free paper.

 2002 Springer-Verlag New York, Inc.

All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed in the United States of America.

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Springer-Verlag New York Berlin Heidelberg

A member of BertelsmannSpringer Science +Business Media GmbH

Preface ix

Software: We designed the text to be independent of any particular software, but have

included appendices in support of the XPPAUT package XPPAUT has been developed

by Bard Ermentrout at the University of Pittsburgh, and it is currently available free

of charge XPPAUT numerically solves and plots the solutions of ordinary differentialequations It also incorporates a numerical bifurcation software and some methods forstochastic equations Versions are currently available for Windows, Linux, and Unixsystems Recent changes in the Macintosh platform (OSX) make it possible to use XPPthere as well Ermentrout has recently published an excellent user’s manual availablethrough SIAM (Ermentrout 2002)

There are a large number of other software packages available that can accomplishmany of the same things as XPPAUT can, such as MATLAB, MapleV, Mathematica, andBerkeley Madonna Programming in C or Fortran is also possible However XPPAUT

is easy to use, requires minimal programming skills, has an excellent online tutorial,and is distributed without charge The aspect of XPPAUT which is available in veryfew other places is the bifurcation software AUTO, originally developed by E.J Doedel.The bifurcation tools in XPPAUT are necessary only for selected problems, so many

of the other packages will suffice for most of the book The the book and web sitecontain code that will reproduce many of the figures in the book As students solve theexercises and replicate the simulations using other packages, we would encourage thesubmission of the code to the editors We will incorporate this code into the web siteand possibly into future editions of the book

There are many people to thank for their help with this project Of course, weare deeply indebted to the contributors, who first completed or wrote from scratch thechapters and then dealt with the numerous revisions necessary to homogenize the book

to a reasonable level Carla Wofsy and Byron Goldstein, as well as Albert Goldbeter,encouraged us to go forward with the project and provided valuable suggestions Wethank Chris Dugaw and David Quinonez for their assistance with typesetting several

of the chapters, and Randy Szeto for his work with the graphic design of the book Wethank James Sneyd for many helpful comments on the manuscript, and also Tim Lewisfor commenting on several of the chapters Carol Lucas generously provided manycorrections for the first half of the text C.F., J.W., and E.M were supported in part bythe Institute of Theoretical Dynamics at UC Davis during some of the preparation ofthe manuscript

We suspect that Joel, for a start, would have thanked Lee Segel, Jim Murray,Leah Edelstein-Keshet and others whose pioneering textbooks in mathematical biol-

ogy certainly informed this one We know that Joel would have thanked many friends

and colleagues for contributing to the true excitement he felt in his “second career”studying biology While we have dedicated this work to the memory of Joel, Joel’sdedication might well have been to his wife, Susan; his daughter, Sarah; his son anddaughter-in-law, Sidney and Noelle; and his grandson, Justin Joel

We hope you enjoy this text, and we look forward to your comments and tions We strongly believe that a textbook such as this might serve to help to develop thefield of computational cell biology by introducing students to the subject This textbook

sugges-Joel Keizer’s thirty years of scientific work set a standard for collaborative research intheoretical chemistry and biology Joel served the University of California at Davis for 28years, as a Professor in the Departments of Chemistry and of Neurobiology, Physiologyand Behavior, and as founder and Director of the Institute for Theoretical Dynamics.Working at the boundary between experiment and theory, Joel built networks of collab-orations and friendships that continue to grow and produce results This book evolvedfrom a textbook that Joel began but was not able to finish The general outline andgoals of the book were laid out by Joel, on the basis of his many years of teaching andresearch in computational cell biology Those of us who helped to finish the project—asauthors and editors—are happy to dedicate our labors to the memory of our friend andcolleague, Joel Edward Keizer All royalties from this book are to be directed to theJoel E Keizer Memorial Fund for collaborative interdisciplinary research in the lifesciences.

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This text is an introduction to dynamical modeling in cell biology It is not meant

as a complete overview of modeling or of particular models in cell biology Rather,

we use selected biological examples to motivate the concepts and techniques used incomputational cell biology This is done through a progression of increasingly morecomplex cellular functions modeled with increasingly complex mathematical and com-putational techniques There are other excellent sources for material on mathematicalcell biology, and so the focus here truly is computer modeling This does not mean thatthere are no mathematical techniques introduced, because some of them are absolutelyvital, but it does mean that much of the mathematics is explained in a more intuitivefashion, while we allow the computer to do most of the work

The target audience for this text is mathematically sophisticated cell biology orneuroscience students or mathematics students who wish to learn about modeling incell biology The ideal class would comprise both biology and applied math students,who might be encouraged to collaborate on exercises or class projects We assume

as little mathematical and biological background as we feel we can get away with,and we proceed fairly slowly The techniques and approaches covered in the first half

of the book will form a basis for some elementary modeling or as a lead in to moreadvanced topics covered in the second half of the book Our goal for this text is toencourage mathematics students to consider collaboration with experimentalists and

to provide students in cell biology and neuroscience with the tools necessary to accessthe modeling literature and appreciate the value of theoretical approaches

The core of this book is a set of notes for a textbook written by Joel Keizer before hisdeath in 1999 In addition to many other accomplishments as a scientist, Joel foundedand directed the Institute of Theoretical Dynamics at the University of California, Davis

It is currently the home of a training program for young scientists studying nonlinear

viii Preface

dynamics in biology As a part of this training program Joel taught a course entitled

“Computational Models of Cellular Signaling,” which covered much of the material inthe first half of this book

Joel took palpable joy from interaction with his colleagues, and in addition to histruly notable accomplishments as a theorist in both chemistry and biology, perhaps hisgreatest skill was his ability to bring diverse people together in successful collaboration

It is in recognition of this gift that Joel’s friends and colleagues have brought this text tocompletion We have expanded the scope, but at the core, you will still find Joel’s hand inthe approach, methodology, and commitment to the interdisciplinary and collaborativenature of the field The royalties from the book will be donated to the Joel E Keizerfoundation at the University of California at Davis, which promotes interdisciplinarycollaboration between mathematics, the physical sciences, and biology

Audience: We have aimed this text at an advanced undergraduate or beginning graduate

audience in either mathematics or biology

Prerequisites: We assume that students have taken full–year courses in calculus and

biology Introductory courses in differential equations and molecular cell biology aredesirable but not absolutely necessary Students with more substantial background ineither biology or mathematics will benefit all the more from this text, especially thesecond half No former programming experience is needed, but a working knowledge ofusing computers will make the learning curve much more pleasant Note that we oftenpoint students to other textbooks and monographs, both because they are importantreferences for later use and because they might be a better source for the material.Instructors may want to have these sources available for students to borrow or consult

Organization: We consider the first six chapters, through intercellular communication,

to be the core of the text They cover the basic elements of compartmental modeling,and they should be accessible to anyone with a minimum background in cell biologyand calculus The remainder of the chapters cover more specialized topics that can beselected from, based on the focus of the course Chapters 7 and 8 introduce spatialmodeling, Chapters 9 and 10 discuss biochemical oscillations and the cell cycle, andChapters 11–13 cover stochastic methods and models These chapters are of varyingdegrees of difficulty

Finally, in the first appendix, some of the mathematical and computational cepts brought up throughout the book are covered in more detail This appendix ismeant to be a reference and a learning tool Sections of it may be integrated into thechapters as the topics are introduced The second appendix contains an introduction tothe XPPAUT ODE package discussed below The final appendix contains psuedocodeversions of the code used to create some of the data figures in the text

con-Internet Resources: This book will have its own web page which will contain a

va-riety of resources We will maintain a list of the inevitable mistakes and typos andwill make available actual code for the figures in the book The web address ishttp://www.compcell.appstate.edu

will be more successful in helping to forge a community if it represents what most of

us agree is necessary to teach beginning students This is only a first step, and we trulylook forward both to input about the material already presented and to suggestionsand contributions of additional material and topics for future editions

New York University

Center for Neural Science

James P Keener

University of Utah

Department of Mathematics

Joel E Keizer

University of California at Davis

Institute of Theoretical Dynamics

Yue-Xian Li

University of British Columbia

Department of Mathematics

Eric S Marland

Appalachian State University

Department of Mathematical Sciences

Alexander Mogilner

University of California at Davis

Department of Mathematics

B´ela Nov ´ak

Budapest University of Technology and

Economics

Department of Agricultural Chemical

Technology

George OsterUniversity of California at Berkeley

Departments of Molecular and Cellular Biologyand ESPM

John E PearsonLos Alamos National Laboratory

Applied Theoretical and Computational Physics

John RinzelNew York University

Center for Neural Science andCourant Institute of Mathematical Sciences

Arthur S ShermanNational Institutes of Health

Mathematical Research BranchNational Institute of Diabetes andDigestive and Kidney Diseases

Gregory D SmithCollege of William and Mary

Department of Applied Science

John J TysonVirginia Polytechnic Instituteand State University

Department of Biology

John M WagnerUniversity of Connecticut Health Center

Center for Biomedical Imaging Technology

Hongyun WangUniversity of California at Santa Cruz

Department of Applied Mathematicsand Statistics

Graphic design by

Randy Szeto

1.1 Scope of Cellular Dynamics 3

1.2 Computational Modeling in Biology 8

1.2.1 Cartoons, Mechanisms, and Models 8

1.2.2 The Role of Computation 9

1.2.3 The Role of Mathematics 10

1.3 A Simple Molecular Switch 11

1.4 Solving and Analyzing Differential Equations 13

1.4.1 Numerical Integration of Differential Equations 15

1.4.2 Introduction to Numerical Packages 18

1.5 Exercises 20

2 Voltage Gated Ionic Currents 21 2.1 Basis of the Ionic Battery 23

2.1.1 The Nernst Potential: Charge Balances Concentration 24

2.1.2 The Resting Membrane Potential 26

2.2 The Membrane Model 27

2.2.1 Equations for Membrane Electrical Behavior 28

2.3 Activation and Inactivation Gates 29

2.3.1 Models of Voltage–Dependent Gating 29

2.3.2 The Voltage Clamp 31

2.4 Interacting Ion Channels: The Morris–Lecar Model 34

2.4.1 Phase Plane Analysis 36

2.4.2 Stability Analysis 38

2.4.3 Why Do Oscillations Occur? 40

2.4.4 Excitability and Action Potentials 43

2.4.5 Type I and Type II Spiking 44

2.5 The Hodgkin–Huxley Model 45

2.6 FitzHugh–Nagumo Class Models 47

2.7 Summary 49

2.8 Exercises 50

3 Transporters and Pumps 53 3.1 Passive Transport 54

3.2 Transporter Rates 57

3.2.1 Algebraic Method 59

3.2.2 Diagrammatic Method 60

3.2.3 Rate of the GLUT Transporter 62

3.3 The Na+/Glucose Cotransporter 65

3.4 SERCA Pumps 70

3.5 Transport Cycles 73

3.6 Exercises 76

4 Fast and Slow Time Scales 77 4.1 The Rapid Equilibrium Approximation 78

4.2 Asymptotic Analysis of Time Scales 82

4.3 Glucose–Dependent Insulin Secretion 83

4.4 Ligand Gated Channels 88

4.5 The Neuromuscular Junction 90

4.6 The Inositol Trisphosphate (IP3) receptor 91

4.7 Michaelis–Menten Kinetics 94

4.8 Exercises 98

5 Whole–Cell Models 101 5.1 Models of ER and PM Calcium Handling 102

5.1.1 Flux Balance Equations with Rapid Buffering 103

5.1.2 Expressions for the Fluxes 106

5.2 Calcium Oscillations in the Bullfrog Sympathetic Ganglion Neuron 107 5.2.1 Ryanodine Receptor Kinetics: The Keizer–Levine Model 108

5.2.2 Bullfrog Sympathetic Ganglion Neuron Closed–Cell Model 111

5.2.3 Bullfrog Sympathetic Ganglion Neuron Open–Cell Model 113

5.3 The Pituitary Gonadotroph 115

5.3.1 The ER Oscillator in a Closed Cell 116

Contents xix

A.2 A Brief Review of Power Series 380

A.3 Linear ODEs 382

A.3.1 Solution of Systems of Linear ODEs 383

A.3.2 Numerical Solutions of ODEs 385

A.3.3 Eigenvalues and Eigenvectors 386

A.4 Phase Plane Analysis 388

A.4.1 Stability of Linear Steady States 390

A.4.2 Stability of a Nonlinear Steady States 392

A.5 Bifurcation Theory 395

A.5.1 Bifurcation at a Zero Eigenvalue 396

A.5.2 Bifurcation at a Pair of Imaginary Eigenvalues 398

A.6 Perturbation Theory 401

A.6.1 Regular Perturbation 401

A.6.2 Resonances 403

A.6.3 Singular Perturbation Theory 405

A.7 Exercises 408

B Solving and Analyzing Dynamical Systems Using XPPAUT 410 B.1 Basics of Solving Ordinary Differential Equations 411

B.1.1 Creating the ODE File 411

B.1.2 Running the Program 412

B.1.3 The Main Window 413

B.1.4 Solving the Equations, Graphing, and Plotting 414

B.1.5 Saving and Printing Plots 416

B.1.6 Changing Parameters and Initial Data 418

B.1.7 Looking at the Numbers: The Data Viewer 419

B.1.8 Saving and Restoring the State of Simulations 420

B.1.9 Important Numerical Parameters 421

B.1.10 Command Summary: The Basics 422

B.2 Phase Planes and Nonlinear Equations 422

B.2.1 Direction Fields 423

B.2.2 Nullclines and Fixed Points 423

B.2.3 Command Summary: Phase Planes and Fixed Points 426

B.3 Bifurcation and Continuation 427

B.3.1 General Steps for Bifurcation Analysis 427

B.3.2 Hopf Bifurcation in the FitzHugh–Nagumo Equations 428

B.3.3 Hints for Computing Complete Bifurcation Diagrams 430

B.4 Partial Differential Equations: The Method of Lines 432

B.5 Stochastic Equations 434

B.5.1 A Simple Brownian Ratchet 434

B.5.2 A Sodium Channel Model 434

B.5.3 A Flashing Ratchet 436

5.3.2 Open–Cell Model with Constant Calcium Influx 122

5.3.3 The Plasma Membrane Oscillator 124

5.3.4 Bursting Driven by the ER in the Full Model 126

5.4 The Pancreatic Beta Cell 128

5.4.1 Chay–Keizer Model 129

5.4.2 Chay–Keizer with an ER 133

5.5 Exercises 136

6 Intercellular Communication 140 6.1 Electrical Coupling and Gap Junctions 141

6.1.1 Synchronization of Two Oscillators 142

6.1.2 Asynchrony Between Oscillators 143

6.1.3 Cell Ensembles, Electrical Coupling Length Scale 144

6.2 Synaptic Transmission Between Neurons 146

6.2.1 Kinetics of Postsynaptic Current 147

6.2.2 Synapses: Excitatory and Inhibitory; Fast and Slow 148

6.3 When Synapses Might (or Might Not) Synchronize Active Cells 150

6.4 Neural Circuits as Computational Devices 153

6.5 Large–Scale Networks 159

6.6 Exercises 165

II Advanced Material 169 7 Spatial Modeling 171 7.1 One-Dimensional Formulation 173

7.1.1 Conservation in One Dimension 173

7.1.2 Fick’s Law of Diffusion 175

7.1.3 Advection 176

7.1.4 Flux of Ions in a Field 177

7.1.5 The Cable Equation 177

7.1.6 Boundary and Initial Conditions 178

7.2 Important Examples with Analytic Solutions 179

7.2.1 Diffusion Through a Membrane 179

7.2.2 Ion Flux Through a Channel 180

7.2.3 Voltage Clamping 181

7.2.4 Diffusion in a Long Dendrite 181

7.2.5 Diffusion into a Capillary 183

7.3 Numerical Solution of the Diffusion Equation 184

7.4 Multidimensional Problems 186

7.4.1 Conservation Law in Multiple Dimensions 186

7.4.2 Fick’s Law in Multiple Dimensions 187

7.4.3 Advection in Multiple Dimensions 188

xvi Contents

7.4.4 Boundary and Initial Conditions for Multiple Dimensions 188

7.4.5 Diffusion in Multiple Dimensions: Symmetry 188

7.5 Traveling Waves in Nonlinear Reaction–Diffusion Equations 189

7.5.1 Traveling Wave Solutions 190

7.5.2 Traveling Wave in the Fitzhugh–Nagumo Equations 192

7.6 Exercises 195

8 Modeling Intracellular Calcium Waves and Sparks 198 8.1 Microfluorometric Measurements 198

8.2 A Model of the Fertilization Calcium Wave 200

8.3 Including Calcium Buffers in Spatial Models 202

8.4 The Effective Diffusion Coefficient 203

8.5 Simulation of a Fertilization Calcium Wave 204

8.6 Simulation of a Traveling Front 204

8.7 Calcium Waves in the Immature Xenopus Oocycte 208

8.8 Simulation of a Traveling Pulse 208

8.9 Simulation of a Kinematic Wave 210

8.10 Spark-Mediated Calcium Waves 213

8.11 The Fire–Diffuse–Fire Model 214

8.12 Modeling Localized Calcium Elevations 220

8.13 Steady-State Localized Calcium Elevations 222

8.13.1 The Steady–State Excess Buffer Approximation (EBA) 224

8.13.2 The Steady–State Rapid Buffer Approximation (RBA) 225

8.13.3 Complementarity of the Steady-State EBA and RBA 226

8.14 Exercises 227

9 Biochemical Oscillations 230 9.1 Biochemical Kinetics and Feedback 232

9.2 Regulatory Enzymes 236

9.3 Two-Component Oscillators Based on Autocatalysis 239

9.3.1 Substrate–Depletion Oscillator 240

9.3.2 Activator–Inhibitor Oscillator 242

9.4 Three-Component Networks Without Autocatalysis 243

9.4.1 Positive Feedback Loop and the Routh–Hurwitz Theorem 244

9.4.2 Negative Feedback Oscillations 244

9.4.3 The Goodwin Oscillator 244

9.5 Time-Delayed Negative Feedback 247

9.5.1 Distributed Time Lag and the Linear Chain Trick 248

9.5.2 Discrete Time Lag 249

9.6 Circadian Rhythms 250

9.7 Exercises 255

10 Cell Cycle Controls 261

10.1 Physiology of the Cell Cycle in Eukaryotes 261

10.2 Molecular Mechanisms of Cell Cycle Control 263

10.3 A Toy Model of Start and Finish 265

10.3.1 Hysteresis in the Interactions Between Cdk and APC 266

10.3.2 Activation of the APC at Anaphase 267

10.4 A Serious Model of the Budding Yeast Cell Cycle 269

10.5 Cell Cycle Controls in Fission Yeast 273

10.6 Checkpoints and Surveillance Mechanisms 276

10.7 Division Controls in Egg Cells 276

10.8 Growth and Division Controls in Metazoans 278

10.9 Spontaneous Limit Cycle or Hysteresis Loop? 279

10.10 Exercises 281

11 Modeling the Stochastic Gating of Ion Channels 285 11.1 Single–Channel Gating and a Two-State Model 285

11.1.1 Modeling Channel Gating as a Markov Process 286

11.1.2 The Transition Probability Matrix 288

11.1.3 Dwell Times 289

11.1.4 Monte Carlo Simulation 290

11.1.5 Simulating Multiple Independent Channels 291

11.1.6 Gillespie’s Method 292

11.2 An Ensemble of Two-State Ion Channels 293

11.2.1 Probability of Finding N Channels in the Open State 293

11.2.2 The Average Number of Open Channels 296

11.2.3 The Variance of the Number of Open Channels 297

11.3 Fluctuations in Macroscopic Currents 298

11.4 Modeling Fluctuations in Macroscopic Currents with Stochastic ODEs 302

11.4.1 Langevin Equation for an Ensemble of Two-State Channels 304

11.4.2 Fokker–Planck Equation for an Ensemble of Two-State Channels 306

11.5 Membrane Voltage Fluctuations 307

11.5.1 Membrane Voltage Fluctuations with an Ensemble of Two-State Channels 309

11.6 Stochasticity and Discreteness in an Excitable Membrane Model 311

11.6.1 Phenomena Induced by Stochasticity and Discreteness 312

11.6.2 The Ensemble Density Approach Applied to the Stochastic Morris–Lecar Model 313

11.6.3 Langevin Formulation for the Stochastic Morris–Lecar Model 314 11.7 Exercises 317

xviii Contents

12.1 Molecular Motions as Stochastic Processes 323

12.1.1 Protein Motion as a Simple Random Walk 323

12.1.2 Polymer Growth 325

12.1.3 Sample Paths of Polymer Growth 327

12.1.4 The Statistical Behavior of Polymer Growth 329

12.2 Modeling Molecular Motions 330

12.2.1 The Langevin Equation 330

12.2.2 Numerical Simulation of the Langevin Equation 332

12.2.3 The Smoluchowski Model 333

12.2.4 First Passage Time 334

12.3 Modeling Chemical Reactions 335

12.4 A Mechanochemical Model 338

12.5 Numerical Simulation of Protein Motion 339

12.5.1 Numerical Algorithm that Preserves Detailed Balance 340

12.5.2 Boundary Conditions 341

12.5.3 Numerical Stability 342

12.5.4 Implicit Discretization 344

12.6 Derivations and Comments 345

12.6.1 The Drag Coefficient 345

12.6.2 The Equipartition Theorem 345

12.6.3 A Numerical Method for the Langevin Equation 346

12.6.4 Some Connections with Thermodynamics 347

12.6.5 Jumping Beans and Entropy 349

12.6.6 Jump Rates 350

12.6.7 Jump Rates at an Absorbing Boundary 351

12.7 Exercises 353

13 Molecular Motors: Examples 354 13.1 Switching in the Bacterial Flagellar Motor 354

13.2 A Motor Driven by a “Flashing Potential” 359

13.3 The Polymerization Ratchet 362

13.4 Simplified Model of the F0Motor 364

13.4.1 The Average Velocity of the Motor in the Limit of Fast Diffusion 366

13.4.2 Brownian Ratchet vs Power Stroke 369

13.4.3 The Average Velocity of the Motor When Chemical Reactions Are as Fast as Diffusion 369

13.5 Other Motor Proteins 374

13.6 Exercises 376

A Qualitative Analysis of Differential Equations 378 A.1 Matrix and Vector Manipulation 379

C Numerical Algorithms 439

P A R T I

Introductory Course

C H A P T E R 1

Dynamic Phenomena in Cells

Christopher P Fall and Joel E Keizer

Over the past several decades, progress in the measurement of rates and interactions ofmolecular and cellular processes has initiated a revolution in our understanding of dy-namic phenomena in cells Spikes or bursts of plasma membrane electrical activity orintracellular signaling via receptors, second messengers, or other networked biochem-ical pathways in single cells, or more complex processes that involve small clusters ofcells, organelles, or groups of neurons, are examples of the complex behaviors that weknow take place on the cellular scale The vast amount of quantitative information un-covered in recent years, leveraged by the intricate mechanisms already shown to exist,results in an array of possibilities that makes it quite hard to evaluate new hypothe-ses on an intuitive basis Using mathematical analysis and computer simulation wecan show that some seemingly reasonable hypotheses are not possible Analysis andsimulations that confirm that a given hypothesis is reasonable can often result in quan-titative predictions for further experimental exploration Rapid advances in computerhardware and software technology combined with pioneering work giving structure tothe interface between mathematics and biology have put the ability to test hypothesesand evaluate mechanisms with simulations within the reach of all cell biologists andneuroscientists

1.1 Scope of Cellular Dynamics

Generally speaking, the phrase dynamic phenomenon refers to any process or

observ-able that changes over time Living cells are inherently dynamic Indeed, to sustain

a depolarizing shock Reprinted from delCastillo and Moore (1959).

the characteristic features of life such as growth, cell division, intercellular nication, movement, and responsiveness to their environment, cells must continuallyextract energy from their surroundings This requires that cells function thermody-namically as open systems that are far from static thermal equilibrium Much energy

commu-is utilized by cells in the maintenance of gradients of ions and metabolites necessaryfor proper function These processes are inherently dynamic due to the continuousmovement of ionic and molecular species across the cell membrane

Electrical activity of excitable cells is a widely studied example of cellular dynamics.The classical behavior of an action potential in the squid giant axon is shown in Figure1.1 This single spike of electrical activity, initiated by a small positive current applied

by an external electrode, propagates as a traveling pulse along the axonal membrane.Hodgkin and Huxley were the first to propose a satisfactory explanation for actionpotentials that incorporated experimental measurements of the response of the squidaxon to depolarizations of the membrane potential We will describe voltage gated ionchannel models in Chapter 2

Membrane transporters allow cells to take up glucose from the blood plasma Cellsthen use glycolytic enzymes to convert energy from carbon and oxygen bonds to phos-phorylate adenosine diphosphate (ADP) and produce the triphosphate ATP ATP, inturn, is utilized to pump Ca2 + and Na+ ions from the cell and K+ions back into thecell, in order to maintain the osmotic balance that helps give red cells the character-istic shape shown in Figure 1.2 ATP is also used to maintain the concentration of2,3-diphosphoglycerate, an intermediary metabolite that regulates the oxygen bindingconformation of hemoglobin The final products of glucose metabolism in red cells arepyruvate and lactate, which move passively out of the cell down a concentration gradi-ent through specific transporters in the plasma membrane Because red cells possessneither a nucleus nor mitochondria, they are not capable of reproduction or energet-ically demanding processes Nonetheless, by continually extracting energy from thetransformation of glucose to lactate, red blood cells maintain the capacity to shuttleoxygen and carbon dioxide between the lungs and the capillaries Remarkably, this

1.1: Scope of Cellular Dynamics 5

cell with its characteristic discoidshape The cell is approximately

5 µm in diameter Reprinted from

Grimes (1980)

efficient biochemical factory is only 5 µm across, with a volume of less than 10−14L

In later Chapter 3 we will discuss models for the transport of glucose across the cellmembrane

Electrical activity and membrane transport are coupled cellular mechanisms perimental measurements of the membrane potential in pancreatic cells have revealedregular bursts of electrical activity corresponding to insulin secretion that is stimulated

Ex-by increases in blood glucose levels These oscillations occur at physiological levels ofglucose, as shown in the microelectrode recordings from a pancreatic beta cell in ananesthetized rat in Figure 1.3 Recent work in vitro has shown that the rapid spikes of

electrical activity, known as action potentials, are caused by rapid influx of Ca2 +fromthe exterior of the beta cell followed by a slower efflux of K+ The periods of rapid

spiking are referred to as active phases of the burst, which are separated by intervals referred to as silent phases A variety of mechanisms have been proposed to explain

bursting behavior, and computer models of bursting were the first to predict that lations of Ca2 +within the cytoplasm should occur in phase with the electrical activity.Oscillations in Ca2 +were recorded for the first time in vitro eight years after they werepredicted by a theoretical model (Chay and Keizer 1983) These oscillations are impor-tant physiologically, because cytoplasmic Ca2 +plays a major role in triggering insulinsecretion This topic will be revisited in Chapter 5

oscil-The control of cellular processes by interlocking molecular mechanisms can alsoproduce spatiotemporal oscillatory Ca2 +signals that are independent of electrical ac-tivity (Lechleiter and Clapham 1992) Figure 1.4 shows the spiral pattern of cytoplasmic

Ca2 +oscillations that occurs when an immature Xenopus leavis egg (an oocyte) is

stim-ulated by a microinjection of inositol 1,4,5-trisphosphate (IP3) IP3is a phospholipidmembrane metabolite that is widely involved in signaling by receptors in the plasmamembrane and that triggers release of Ca2 +from the endoplasmic reticulum (ER) The

Figure 1.3 Periodic bursts of electrical activity recorded in vivo from a pancreatic beta cellfrom the intact pancreas of an anesthetized rat Reprinted from Sanchez-Andres et al (1995)

ER is an intracellular compartment that functions, among other purposes, as a store for

Ca2 + The ER maintains an internal Ca2 +concentration ([Ca2 +]ER) that is comparable

to that of the external medium (ca 5 mM), whereas the cytoplasmic Ca2 +concentration([Ca2 +]i) is typically 1000-fold smaller The spiral waves of Ca2 +in Xenopus oocytes can

be explained quantitatively by kinetic models of the feedback mechanisms ble for uptake and release of Ca2 + from the ER Simple models of regenerative Ca2 +

responsi-release that are solved on a spatial domain provide insight into the processes of self

organization that result in spiral waves (Winfree 1987) These models are discussed in

Chapter 8

Circadian rhythms are regular changes in cellular processes that have a period

of about 24 hours (from the Latin circa, about, and dies, day) and represent another

dynamical phenomenon that is widely observed in cells A great deal about the nisms of circadian rhythms has been uncovered in recent years, and circadian biologyoffers a rich source of unsolved modeling problems Internal clocks provide an or-ganism with the ability to predict changes in the environment For example, floweropening and insect-egg hatching occur in advance of the rising sun (Winfree 1987)

mecha-Cell division in Euglenids may also synchronize to light–dark cycles, as shown in

Fig-ure 1.5 The dark/light bands correspond to periods of absence and presence of lightthat simulate the normal dark/light cycle during a day As shown in Figure 1.5, the

growth rate of Euglena is temperature dependent, and cell division sychronizes to a

24–hour dark/light cycle only when the temperature is in the range found in its naturalenvironment At this temperature the population doubling time is close to 24 hours

Recent experiments with the fruit fly Drosophila and other organisms suggest that

cir-cadian rhythms like this are controlled by oscillations in gene transcription Furtherconsideration of circadian rhythms will be given in Chapter 9

1.1: Scope of Cellular Dynamics 7

Figure 1.4 A spiral wave of Ca2 +ions detected as the bright fluorescence from an indicator

dye after microinjection of IP3 into an immature frog egg, the Xenopus laevis oocyte Provided

by Drs James Lechleiter and Patricia Camacho, University of Texas Health Sciences Center

The cell division cycle is the process by which a cell grows and divides into twodaughter cells that can repeat the process The eukaryotic cell cycle consists of a regularsequence of events as shown in Figure 1.6: chromosome replication during a restrictedperiod of the cycle (S phase), chromosome segregation during metaphase and anaphase

(M phase, or mitosis), and finally cytokinesis, in which two daughter cells separate.

This cycle involves a cascade of molecular events that center on the proteins Cdc2 andcyclin, which make up a complex known as M phase promoting factor, or MPF Thiscomplex has been shown to oscillate in synchrony with cell division and to be regulated

by a series of phosphorylation and dephosphorylation reactions Related dynamical

changes occur during meiosis, in which germ line cells produce eggs and sperm We

will discuss models of the cell cycle progression in Chapter 10

After DNA replication is complete, each chromosome consists of two “sister matids,” which must be separated during mitosis so that each daugher nucleus getsone and only one copy of each chromosome Segregation of sister chromatids duringmitosis is another complex dynamical process that involves self-organizing structures

chro-in the cell that pull sister chromatids apart This wonderfully coordchro-inated dynamicalbehavior is just one of many examples of motile cellular processes Other importantexamples include muscle contraction, cell movement, and projections of cell mem-

brane called pseudopodia Molecular motors will discussed at length in Chapter 12 and

Chapter 13

Figure 1.5 Population growth of a mutant

Euglena strain shows log growth at 25◦C dependent of the light/dark cycle, indicated

in-by the alternating light/dark bands on thetime axis At 19◦C a circadian (approximatly

24 hour) growth rhythm develops that trains with the light/dark cycle Reprintedfrom Edwards, Jr (1988)

en-1.2 Computational Modeling in Biology

Even the simplest of the dynamic phenomena described in the previous section areexceedingly complex, and computer models have proven to be an important tool inhelping to dissect the molecular processes that control their evolution in time In thephysical sciences, theoretical methods in combination with experimental measure-ments have for many years provided rich insights into dynamical phenomena Theabundance of quantitative experimental data now available at the cellular level hasopened the door to similar collaborations in neurobiology and cell physiology

1.2.1 Cartoons, Mechanisms, and Models

The interplay of experiment, theory, and computation follows a conceptual frameworksimilar to that which has proven successful in the physical sciences:

1 Taking clues from experimental work, the first step is to sort through possiblemolecular mechanisms and focus on the most plausible ones In most cases, thisstep requires close consultation with experimentalists working on the problem

2 The selection of mechanisms defines the basis for a schematic representation, or

cartoon, that depicts the overall model To be useful, the cartoon should be explicit

enough to be translated into a series of elementary steps representing the individualmechanisms

3 Next, the basic laws of physics and chemistry can be used to translate theelementary steps of the mechanism into mathematical expressions

1.2: Computational Modeling in Biology 9

ANAPHASE METAPHASE

Spindle formation Chromosome

condensation

Spindle pole body duplication

DNA replication

Growth

Cytokinesis

Nuclear divison

Chromosome segregation

START

Figure 1.6 The cell division cycle in fission yeast Spindle formation initiates metaphase,where condensed chromosomes pair up for segregation into daughter cells The cycle iscompleted when cytokinesis cleaves the dividing cell

4 These expressions are then combined into time dependent differential equationsthat quantify the changes described by the whole model

5 Finally, careful study of the differential equations reveals whether or not thecartoon is a successful model of the biological system

The challenge of the theorist in biology then becomes similar to that in astrophysics

or quantum mechanics: to analyze the equations, simplify them if possible, solve them,and, most importantly, make predictions that can be tested by further experiment.Further experiments may uncover inconsistencies in a model that will require changes.The process that we have outlined above and will revisit in later chapters is an iterativecycle of ever improving approximation where the mathematical or computer modelplays the role of a quantitative hypothesis

1.2.2 The Role of Computation

Advances in computer hardware and numerical analysis have made the solution of plicated systems of ordinary differential equations fast, accurate, and relatively easy.Indeed, the role of computation is critical because the differential equations describ-ing biological processes nearly always involve control mechanisms that have nonlinearcomponents Simple linear differential equations often can be solved analytically, whichmeans that we can obtain an exact solution using traditional mathematical methods.Nonlinearities often make it difficult or impossible to obtain an exact solution; how-

com-ever, we can obtain quite good estimates using numerical methods implemented oncomputers Spatial variation is often an important feature in cellular mechanisms, so

one is confronted with analyzing and solving spatially explicit partial differential

equa-tions, which can be still more complicated and less analytically tractable than ordinary

differential equations

Computer models permit one to test conditions that may at present be difficult toattain in the laboratory or that simply have not yet been examined by experimentalists

Each numerical solution of the differential equations can therefore provide a

simula-tion of a real or potential laboratory experiment These simulasimula-tions can be used to help

assess parameters, such as diffusion constants or kinetic constants, that may be ficult to measure experimentally Numerical simulations can test how intervention bypharmacological agents might affect a process With simulations one can test specifichypotheses about the role of individual mechanistic components or make predictionsthat can be tested in the laboratory Often the most important result of a simulation

dif-is negative: A well–crafted model can rule out a particular mechandif-ism as a possibleexplanation for experimental observations

1.2.3 The Role of Mathematics

The scope of mathematical techniques employed to investigate problems in matical biology spans almost all of applied mathematics The modeling of processes

mathe-is dmathe-iscussed in detail here, but only the basics of the mathematics and the elementarytools for the analysis of these models are introduced Rigorous mathematics plays atleast three important roles in the computational modeling of cell biology One role is

in the development of the techniques and algorithms that make up the tools of

numeri-cal analysis In its essence, the computation of solutions to mathematinumeri-cal problems on

computers is fundamentally a process of estimation, and the accuracy and efficiency ofthese methods of estimation are the subjects of much study We will introduce briefly afew ideas from numerical analysis at the end of this chapter and in appropriate placesthroughout the rest of this book

The process of developing model mechanisms that we described above is alsofundamentally a process of approximation due to the simplifications that must be in-troduced to produce a useful model Not only must these simplifications make sense interms of the physical process being studied, but they must also be valid from a mathe-matical standpoint We will learn the basics about more mathematical concepts such

as the reduction of scale and stochastic methods in later chapters

It is one thing to solve the differential equations that result from the tion of a model, but another thing to understand why a model behaves as it does.Mathematicians have developed techniques and tools for the analysis of systems of dif-ferential equations that describe complex interrelated processes These tools reveal thestructure, properties, and dynamical behavior of the system much as anatomical, phys-iological, and molecular biological techniques reveal the physical basis of the model

formula-In particular, such analysis reveals details about behaviors in a model such as the

os-1.3: A Simple Molecular Switch 11

Figure 1.7 Mechanistic cartoon of a gated ionic channelshowing an aqueous pore that is selective to particulartypes of ions The portion of the transmembrane proteinthat forms the “gate” is sensitive to membrane potential,allowing the pore to be in an open or closed state

cillations and other complex behaviors that often motivate study of such biologicalphenomena These techniques are covered in several chapters and in the appendices.Rigorous analysis of complicated differential equations requires specialized train-ing, because there are many subtleties that are appreciated only with experience.Similarly, choosing proper numerical methods and selecting valid simplifications re-quires caution While the creation and manipulation of simple models is within thereach of all cell biologists, the careful scientist will seek collaborations with experi-enced mathematicians, particularly for the valid simplification of complicated modelsinto more tractable ones In the middle ground between established disciplines such

as biology and mathematics, fruitful scientific work can be done, and all parties gainvaluable insight from the interdisciplinary experience

1.3 A Simple Molecular Switch

In this section we illustrate, with a simple model of channel conformation, the kinds ofphenomena that are investigated in detail in subsequent chapters We introduce some

of the methods underlying the analysis of these models and also try to demonstrate thebasic conceptual modeling framework utilized throughout the book

We begin with a simple channel because it is an intuitively clear example of sition between different molecular states corresponding to different conformations of

tran-a mtran-acromolecule Let us be cletran-ar thtran-at we tran-are modeling only proteins thtran-at tran-are ing between an “open” state and a “closed” state and nothing more at this point Thesimplest cartoon of gating is a channel with two states, one with the pore open andthe other with it closed, corresponding to the mechanism shown in Figure 1.7 Thiskinetic “cartoon” is easily translated into a conventional kinetic model of the sort oftenemployed in biochemistry

switch-The model takes the form of the diagram in (1.1) Diagrams like this, which will

be used extensively in this book, represent molecular states or entities by symbols andtransitions between states by solid lines or arrows:

C

k

The C in (1.1) corresponds to the closed state, the O corresponds to the open state of the

channel, and the arrows represent elementary molecular processes These states resent a complex set of underlying molecular conformations The transitions between

rep-C and O are unimolecular processes because they involve only the channel molecule (bimolecular processes will be introduced in Chapter 3) An important aspect of tran-

sitions between molecular states is that they are reversible, which is a consequence ofmicroscopic reversibility of molecular processes

Using special techniques, transitions between closed and open states can be sured for single channel molecules However, here the focus is on the average changefor a collection of channels Because it is not unusual to have several thousand chan-nels of a given type in the plasma membrane of a cell, the average behavior of the entireensemble of channels is often what determines the cellular dynamics

mea-The rate of an elementary process in a kinetic diagram is determined by the

so-called law of mass action (Despite the name, mass action is not technically a physical

law, but rather is a constitutive relation that holds as a very good approximation for anywell–mixed system.) This “law,” which dates back to the early studies of chemical kinet-ics, states that the rate of a process is proportional to the product of the concentrations

of the molecular species involved in the process If we define k+as the proportionality

constant or the rate constant, the rate of the transition from state C to state O is given

by J+ k+[C], where the square brackets denote concentration, with [C] representing

the concentration of channel molecules in state C In this case k+is unimolecular, withpractical units of s−1 Similarly, the rate of the reverse reaction, C ← O, is given by

J− k−[O] with the unimolecular rate constant k−

To translate the mechanism in (1.1) into an equation, the law of mass action is

applied to the concentration of channels in states C and O For cellular mechanisms,

a variety of measures of “concentration” can be used For example, if the channels are

in intact cells, concentration is often expressed in terms of total cell volume Anothermeasure in common use involves total weight of protein in a sample The total number

of transporters is useful for single cells Here we choose the latter to define tion so that [O] f O  N O /N , where f Owill refer to the fraction of open channels, and

concentra-N and N O the total number of channels and open channels, respectively (similarly, f C

and N Crefer to the closed state)

Because the kinetic model involves only interconversion of channel states, the totalnumber of channels should be preserved This introduces the idea of a conservation

law, N C + N O  N, which states mathematically that channels are neither created

nor destroyed Using conservation relations, one of the dependent variables can be

eliminated because N C  N − N O The differential equation for N C therefore becomesredundant, and the number of differential equations to be solved is reduced to only

one along with the algebraic equation for N C The fraction of channels in the closedstate is therefore 1− f O

Having established the correspondence of the diagram with rate expressions, it iseasy to write down the differential equations that the diagram represents To do soone must keep track of the change that each elementary process in the diagram makes

1.4: Solving and Analyzing Differential Equations 13

for each state, which we refer to as a flux Thus the process connecting states O and C causes a loss of state O in the reverse direction and again in the forward direction These

small whole numbers that correspond to losses or gains of a state (e.g., −1 for state

C in the process C → O) are called the stoichiometric coefficients for the mechanism.

Using the coeficients in conjunction with the kinetic diagram, the ordinary differentialequations follow for the rate of change in the states:

1.4 Solving and Analyzing Differential Equations

Many students have worked with differential equations in their studies of the physicalsciences or elementary mathematics, and they may have been introduced to solution

techniques explicitly in an advanced calculus course In general, the differential tions that arise for the rate of change in cellular properties will be complicated anddifficult or impossible to solve exactly using analytical techniques One example thatcan be solved is the simple linear equation

It can be verified that (1.7) is indeed a solution of (1.6) by differentiating the solution

with respect to t, thereby recovering our original equation

at what value it begins Thus there is a family of solutions to a differential equation,

and the correct one is chosen by specifying an initial condition This is an important

concept that is particularly relevant to the numerical solution of differential equations

We choose a particular solution to (1.6) from the general solution given by (1.7) by adjusting the constant C such that the initial condition X  X(0) is satisfied at time

A deep understanding of the rules of differentiation and integration from calculus

is not required for the level of modeling and mathematics that will be encountered inthe first half of this book The exponential function is encountered frequently, and so areview of its properties as given in elementary calculus textbooks is advisable It is alsohelpful when analyzing experimental literature to understand the difference between

the time constant τ and the half-time t 1/2 in the context of the exponential function.The half-time is the time at which the value of the function decays to 12 of the inititalvalue, or the solution of

X (0)e −t/τ  X(0)

1.4: Solving and Analyzing Differential Equations 15

Note that these principles apply to growing as well as decaying functions

The solution X(t) always “relaxes” or decays to zero at long times (Note that to mathematicians an equation decays to steady state whether it approaches a value

greater or smaller than the initial value.) An equation related to the simple exponentialdecay equation given in (1.6) is the exponential approach to a steady state other thanzero The single channel model given in (1.5) describes exponential decay (or growth) to

a steady state fraction of open channels, f∞ We rewrite it here to aid the presentation:

1.4.1 Numerical Integration of Differential Equations

Even if (1.14) were more complicated and could not be solved exactly, a numericalapproximation could still be calculated The simplest and perhaps the oldest method

of numerical solution goes back to the mathematician Euler and is easy to

under-stand The method is called the forward Euler method and it is a prototype for all other

numer-time point The solution to (1.6) with τ  2 isshown in solid, the linear approximation using

the derivative at t  0 (−0.5) is shown in dots,

and the difference between these two curves

is shown in dashes Note that by t  2, themagnitude of error between the actual and ap-proximate functions is equal to the value of theactual function

methods of solving ODEs numerically Consider an approximation to the derivative in(1.6)

dX

dt ≈ X

t  X (t + t) − X(t)

where X and t are small, but not infinitesimal like the differentials dX and dt If

this approximation to the derivative is substituted into the differential equation, the

resulting equation can be solved for X(t + t), giving

The smaller t is, the better the Euler approximation of the derivative will be Also,

because the Euler approximation gives a piecewise linear estimate of the solution, the

further from linear the problem is, the smaller t must become to give an accurate

solution (see Figure 1.9) The essence of numerical integration is that we start at some

value and crawl along the solution in increments of t by estimating the change over that interval If t is very small, our estimate of the rate of change is good and our

solution is accurate, but it may take a very long time to compute the solution This

is termed computationally expensive, because it either requires a faster computer or a

longer time to run

Two solutions to (1.19) obtained by integrating the equation using the Euler method

are shown in Figure 1.10A The time step was chosen to be dt  0.03, and two different initial conditions were used, f O(0) 1 and f O(0) 0 Independent of the initial condi-

tion, f O (t) relaxes to its steady state value f∞ 0.5 It corresponds to the point where the

rate vanishes, as can be seen graphically in Figure 1.10A The rate at which the steady

state is approached depends on the value of τ, which is 3 in these simulations.

1.4: Solving and Analyzing Differential Equations 17

A

B

0.2 0.1 0.0 0.1 0.2

0.2 0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0

f ∞

Figure 1.10 (A) The exponential decay of the open fraction of channels Initial conditions ateither 1 or 0 both decay to the steady–state value of 0.5 (B) The effect of the step size in the

Euler method for the simulation in panel A starting with the initial condition f O(0)  1 The

exact solution, f O (t )  0.5(1 + exp(−t/3)), is given by the full line (C) Plot of the rate of change

of n as a function of n (D) Phase portrait with the arrows representing the direction and relative magnitude of the rate for each value of f O All the arrows point toward the steady state, f∞ 0.5.

The solution to the equation in the Euler method depends on step size as shown inFigure 1.10B Only step sizes that are more than an order of magnitude smaller than the

value of τ do a reasonable job of approximating the exact exponential solution, which for the parameter values used is f O (t)  0.5(1 + exp(−t/3)) Unreasonably large step sizes like t 6 give approximations that are not even close to the exact solution In

fact, the numerical method has become unstable, and the computed solution oscillates

around the true solution

There are many other methods of numerical integration that give better imations to the derivative These methods are generally more complicated, but have

approx-fewer restrictions on t These more complicated methods also address some other portant problems such as numerical stability There are many fine texts on numerical

im-analysis that discuss these issues and explain the various advantages and disadvantages

of each method As mentioned above, however, the best way to ensure an optimal andvalid means of solution is to collaborate with a mathematician who has experience inscientific computation or numerical analysis

There is a different way to plot the results of solving the differential equation thatfrequently gives insight into the properties of the solution This is demonstrated in

Figure 1.10C, where the function df O /dt  −(f O − f∞)/τ is plotted versus the value of f O

for the two initial conditions in Figure 1.10A Since f Ois restricted on physical grounds

to be between 0 and 1, the plot shows that f∞ is the unique steady state by making it

clear that when f O > 0.5, n decreases with time because df O /dt < 0, and when f O < 0.5,

f O (t) increases with time.

To further emphasize this concept, arrows in Figure 1.10D show the direction in

which f O is changing This type of plot is called a phase portrait, in one dimension.

Phase portraits are particularly useful for analyzing ODEs with two variables, where

they are typically called phase plane diagrams Phase plane diagrams are discussed in

Appendix A Because phase portrait diagrams will be used extensively in the remainder

of the text, it would be a useful digression to review that material now

1.4.2 Introduction to Numerical Packages

While it is important to understand the limitations of whichever numerical algorithm

is used for the solution of a problem, fortunately it is not necessary to face the task ofimplementing these algorithms on a computer from scratch Several excellent softwareprograms have been developed that not only solve ODEs, but represent solutions graph-ically and allow their dynamical properties to be analyzed These packages include verysophisticated commercial mathematical packages such as Matlab and Mathematica,which can often be obtained at reduced cost in the form of student versions There arealso myriad university produced packages such as Berkeley Madonna that are designed

to solve ODEs A new direction for computational cell biology is the creation of severalhighly sophisticated packages such as The Virtual Cell, which is an integrated databaseand computational system expressly designed for cell biology modeling

Among the best for our purposes here is a public domain package, XPPAUT, thathas been developed by Bard Ermentrout at the University of Pittsburgh (Ermentrout

2002) The name of the program evolved from a DOS version that was called

Phase-Plane, refering to the program’s ability to carry out phase plane analysis A version that

ran in X-windows under Unix or linux was then developed and was called X-PhasePlane(or XPP for short) Finally, when the automatic bifurcation tool AUTO developed by

E Doedel, was added it became X-PhasePlane-Auto, or XPPAUT XPPAUT is an lent tool for solving and analyzing ordinary differential equations, one that provides

excel-1.4: Solving and Analyzing Differential Equations 19

very sophisticated tools without automating the process too much to be useful forour didactic objectives Moreover, XPPAUT is free and available for both Unix/Linux,Windows, and MacOSX We adopt it as the basic software program for use with thistext

Suggestions for Further Reading

At the end of each chapter will be listed several sources for further reading, togetherwith short descriptions Because Chapter 1 serves as an introduction to the whole book,

we have listed here several sources that might serve as companions to the whole book.Because of their applicability to more than one chapter, those listed here may be listedagain later

• Modeling Dynamic Phenomena in Molecular and Cellular Biology, Lee Segel A great

place to start for a more mathematical treatment of some of the contents of thisbook (Segel 1984)

• Mathematical Models in Biology, Leah Edelstein-Keshet A classic introductory

textbook for general mathematical biology, and a good source for a differentperspective on topics such as time scales, phase plane analysis, and elementarynumerical analysis as applied to biological problems (Edelstein-Keshet 1988)

• Understanding Nonlinear Dynamics, Daniel Kaplan and Leon Glass, and

Nonlin-ear Dynamics and Chaos, Steven Strogatz Extremely readable entry–level books

on nonlinear dynamics, including sections on chaos, fractals, and data analysis(Kaplan and Glass 1995; Strogatz 1994)

• Mathematical Physiology, James Keener and James Sneyd Keener and Sneyd treat

many of the topics presented in this book from a more analytic perspective asopposed to the computational focus presented here (Keener and Sneyd 1998)

• Computer Methods for Ordinary Differential Equations and Differential-Algebraic

Equations, U.M Acher and L.R Petzold (Asher and Petzold 1998).

• Cellular Biophysics, Volumes 1 and 2, Thomas Weiss These two volumes cover in

more detail the biophysics of transport processes and electrical properties in cells(Weiss 1996)

• Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for

Researchers and Students, by Bard Ermentrout A complete user’s manual for the

public domain ODE package XPP (Ermentrout 2002)

• Mathematical Biology, James Murray While not especially didactic, this volume is

recognized as an essential handbook describing models throughout mathematicalbiology (Murray 1989)